Einstein Metrics, Harmonic Forms, and Symplectic Four-Manifolds

TL;DR

This paper characterizes a specific region in the space of Riemannian metrics on certain 4-manifolds, showing it contains all known Einstein metrics and no others, thus identifying a unique component in the Einstein metric moduli space.

Contribution

It identifies a particular subset of metrics on Del Pezzo surfaces that includes all known Einstein metrics and proves no additional Einstein metrics exist in that region.

Findings

The region contains all known Einstein metrics on the manifold.

No other Einstein metrics exist within this region.

This region forms a unique connected component in the moduli space.

Abstract

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(\omega , \omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.